To add the mixed numbers \( 3 \frac{5}{7} \) and \( 7 \frac{1}{2} \), follow these steps:
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Convert the mixed numbers to improper fractions.
For \( 3 \frac{5}{7} \): \[ 3 \frac{5}{7} = \frac{3 \times 7 + 5}{7} = \frac{21 + 5}{7} = \frac{26}{7} \]
For \( 7 \frac{1}{2} \): \[ 7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2} \]
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Find a common denominator.
The least common multiple of 7 and 2 is 14.
Convert \( \frac{26}{7} \) to an equivalent fraction with a denominator of 14: \[ \frac{26}{7} = \frac{26 \times 2}{7 \times 2} = \frac{52}{14} \]
Convert \( \frac{15}{2} \) to an equivalent fraction with a denominator of 14: \[ \frac{15}{2} = \frac{15 \times 7}{2 \times 7} = \frac{105}{14} \]
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Add the two fractions: \[ \frac{52}{14} + \frac{105}{14} = \frac{52 + 105}{14} = \frac{157}{14} \]
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Convert the improper fraction back to a mixed number. Divide 157 by 14:
- \( 14 \) goes into \( 157 \) eleven times (since \( 14 \times 11 = 154 \)).
- The remainder is \( 157 - 154 = 3 \).
Thus, \[ \frac{157}{14} = 11 \frac{3}{14} \]
Therefore, the sum of \( 3 \frac{5}{7} + 7 \frac{1}{2} \) is \( 11 \frac{3}{14} \).